1. HILBERT TRANSFORM
Fourier, Laplace, and z-transforms change from the time-domain representation of a signal to the frequency-domain representation of the signal
The resulting two signals are equivalent representations of the same signal in terms of time or frequency
In contrast, The Hilbert transform does not involve a change of domain, unlike many other transforms

2. HILBERT TRANSFORM
Strictly speaking, the Hilbert transform is not a transform in this sense
First, the result of a Hilbert transform is not equivalent to the original signal, rather it is a completely different signal
Second, the Hilbert transform does not involve a domain change, i.e., the Hilbert transform of a signal x(t) is another signal denoted by in the same domain (i.e.,time domain)

3. HILBERT TRANSFORM
The Hilbert transform of a signal x(t) is a signal whose frequency components lag the frequency components of x(t) by 90
has exactly the same frequency components present in x(t) with the same amplitude–except there is a 90 phase delay
The Hilbert transform of x(t) = Acos(2f0t + ) is Acos(2f0t +  - 90) = Asin(2f0t + )

4. HILBERT TRANSFORM A delay of /2 at all frequencies
ej2f0t will become
e-j2f0t will become
At positive frequencies, the spectrum of the signal is multiplied by -j
At negative frequencies, it is multiplied by +j
This is equivalent to saying that the spectrum (Fourier transform) of the signal is multiplied by
-jsgn(f).

5. HILBERT TRANSFORM
Assume that x(t) is real and has no DC component : X(f)|f=0 = 0,
then
The operation of the Hilbert transform is equivalent to a convolution, i.e., filtering

6. Example 2.6.1
Determine the Hilbert transform of the signal x(t) = 2sinc(2t)
Solution
We use the frequency-domain approach . Using the scaling property of the Fourier transform, we have
In this expression, the first term contains all the negative frequencies and the second term contains all the positive frequencies
To obtain the frequency-domain representation of the Hilbert transform of x(t), we use the relation = -jsgn(f)F[x(t)], which results in
Taking the inverse Fourier transform, we have

7. HILBERT TRANSFORM
Obviously performing the Hilbert transform on a signal is equivalent to a 90 phase shift in all its frequency components
Therefore, the only change that the Hilbert transform performs on a signal is changing its phase
The amplitude of the frequency components of the signal do not change by performing the Hilbert-transform
On the other hand, since performing the Hilbert transform changes cosines into sines, the Hilbert transform of a signal x(t) is orthogonal to x(t)
Also, since the Hilbert transform introduces a 90 phase shift, carrying it out twice causes a 180 phase shift, which can cause a sign reversal of the original signal

8. HILBERT TRANSFORM - ITS PROPERTIES
Evenness and Oddness
The Hilbert transform of an even signal is odd, and the Hilbert transform of an odd signal is even
Proof
If x(t) is even, then X(f) is a real and even function
Therefore, -jsgn(f)X(f) is an imaginary and odd function
Hence, its inverse Fourier transform will be odd
If x(t) is odd, then X(f) is imaginary and odd
Thus -jsgn(f)X(f) is real and even
Therefore, is even

9. HILBERT TRANSFORM - ITS PROPERTIES
Sign Reversal
Applying the Hilbert-transform operation to a signal twice causes a sign reversal of the signal, i.e.,
Proof
X( f ) does not contain any impulses at the origin

10. HILBERT TRANSFORM - ITS PROPERTIES
Energy
The energy content of a signal is equal to the energy content of its Hilbert transform
Proof
Using Rayleigh's theorem of the Fourier transform,
Using the fact that |-jsgn(f)|2 = 1 except for f = 0, and the fact that X(f) does not contain any impulses at the origin completes the proof

11. HILBERT TRANSFORM - ITS PROPERTIES
Orthogonality
The signal x(t) and its Hilbert transform are orthogonal
Proof
Using Parseval's theorem of the Fourier transform, we obtain
In the last step, we have used the fact that X(f) is Hermitian; | X(f)|2 is even

12. Ideal filters Ideal lowpass filter
An LTI system that can pass all frequencies less than some W and rejects all frequencies beyond W
W is the bandwidth of the filter.

13. Ideal filters Ideal highpass filter
There is unity outside the interval -W  f W and zero inside
Ideal bandpass filter
have a frequency response that is unity in some interval
–W1  |f| W2 and zero otherwise
The bandwidth of the filter is W2 – W1

14. Nonideal filters – 3dB bandwidth
For nonideal filters, the bandwidth is usually defined as the band of frequencies at which the power of the filter is at least half of the maximum power
This bandwidth is usually called the 3 dB bandwidth of the filter, because reducing the power by a factor of two is equivalent to decreasing it by 3 dB on the logarithmic scale

15. LOWPASS AND BANDPASS SIGNALS
Lowpass signal
A signal in which the spectrum (frequency content) of the signal is located around the zero frequency
Bandpass signal
A signal with a spectrum far from the zero frequency
The frequency spectrum of a bandpass signal is usually located around a frequency fc
fc is much higher than the bandwidth of the signal